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Optimizing topological states
A topological state is also an important form of wave characteristics in finite structures. We now turn to
focus on some recent design works on topological states. Generally speaking, topological invariants can
characterize topological properties of structures, but their definition and calculation are often difficult. In
essence, topological properties certainly exist in structural features, so exploring topological properties from
actual structures instead of relying on topological invariants is another idea for topological classification.
Long et al. demonstrated an unsupervised clustering algorithm for extracting topological features of
[121]
phononic crystals, thereby classifying topological properties . He et al. achieved the inverse design of
phononic crystal thin plates with anticipated bandgap width and topological property based on MLP, as
[122]
shown in Figure 3C . By designing two units with a broadband common bandgap, they constructed a
highly robust localized edge state for bending wave transmission. This group subsequently proposed using
TNN to achieve the inverse design of phononic beams from topological properties to structure . The
[105]
topological properties of the bandgap were characterized by the reflection phase, and the interface states of
one-dimensional phononic beams were predicted and constructed using TNN. Afterward,
Muhammad et al. also completed a similar work . Du et al. realized the inverse design of Valley Hall
[123]
acoustic topological insulator by combining MLP and GA, as shown in Figure 3D . Specifically, they first
[124]
trained regression neural networks and classification neural networks for predicting bandgap and
topological properties, respectively. Then, two neural networks are put into the optimization process of GA
to obtain two structures with opposite topological properties under a common bandgap for constructing
edge states.

The application of ML in Hermitian systems mentioned above is still in the initial stage, and more
achievements need to be further expanded. At the same time, we have also found that ML has recently made
some attempts in non-Hermitian systems. Yu et al. used diffusion maps to unsupervised manifold learning
of topological phases in non-Hermitian systems . Different from the unsupervised method, there are also
[125]
some works that demonstrate training ANNs for supervised prediction of non-Hermite topological
invariants [126-128] . The essential difference between unsupervised and supervised is that the former does not
need labels and directly extracts topological invariant from the on-site elements of the model, while the
latter relies on the calculated topological invariant as labels to construct data sets. In non-Hermitian
systems, an exception point (EP) is an important feature that represents the critical point at which the
system transitions from a real eigen-spectrum to a complex eigen-spectrum . In the latest work, Reja et al.
[129]
[130]
introduced neural networks for the characterization of EP . They proposed a method called summed
phase rigidity (SPR) to characterize the order of EPs in different models. Then, they trained MLP models to
realize the prediction of EPs for two-site and four-site gain and loss models.

Design of static characteristics in mechanical meta-structures
Mechanical meta-structures have become an emerging growth point in the field of ML-enabling design due
to their extreme statics performance. Combined with ML, meta-structures with excellent mechanical
properties can be obtained through design optimization by adding, deleting, or changing. Table 3 provides a
brief overview of ML for the design of static characteristics in mechanical meta-structures.

A lot of work has been carried out around the 2D mechanical meta-structures. These structures are usually
designed and optimized on a plane to obtain specific shapes or material compositions with specific
mechanical properties. CNN, as a high-quality model for image feature extraction, is widely used in the
design of 2D mechanical meta-structures. Gu et al. proposed a self-learning CNN model to search for high-
performance hierarchical mechanical structures . This model can continuously learn patterns from high-
[131]
performance structures, ultimately achieving design results superior to the training set. Hanakata et al.
reported a design study on stretchable graphene kirigami, as shown in Figure 4A . The cutting density and
[132]